Studies On Portfolio Selection Problems With Different Risk Measuresand Trading Constraints

As an important part of modern finance theory, investment science focuses on how to help investors make reasonable investment decisions and manage risk under the risk-return framework. In1952, Markowitz proposed the famous mean-variance portfolio selection model, which has been regarded as the foundation of modern investment science. Over the past six decades, many researchers have made great efforts to extend mean-variance model. Different risk measures have been proposed from different perspectives. The advances and development of modern investment science provide efficient tools for risk management and the-oretical guide for new financial innovations.Markowitz’s mean-variance model, however, has some evident drawbacks. The variance measures the overall risk of portfolio, while neglecting the marginal contributions of individual assets or factors. In practice, classical mean-variance model is merely used as a theoretical model, and it does not take into account of the practical features, such as cardinality constraint and minimum buy-in threshold. Meanwhile, for the portfolios with asymmetry returns, mean and variance cannot grasp all the characteristics of portfolio returns. All of these drawbacks undermine the practical applications of mean-variance model.We focus in this thesis on the portfolio selection problems based on differ-ent risk measures and trading constraints in order to narrow the gap between investment theory and financial practice. The main contributions of the thesis are described as follows.The adoption of an appropriate factor model enables us to pin point the hidden forces that drive the movement of the market and contribution to the systematic risk of the market. We introduce the concept of factor risk to measure individual factor’s contribution to the overall risk, and construct the portfolio se-lection model with factor risk control. The resulting model can be reduced to an quadratically constrained quadratic program. By exploiting the special features of the model structure, we propose a special branch-and-bound algorithm which employs the second-order cone relaxations as its bounding procedure. Numerical test indicates that our algorithm can optimize the model efficiently. Historical data from Hong Kong stock market are employed to analyze the empirical per-formance. Analysis results show that the portfolio selection model with factor risk control can help to generate robust portfolio and circumvent the influence of unpromising factors.The efficiency of portfolio selection models depends on the accuracy of model parameters. Since estimation error cannot be avoided in parameter estimation, we analyze the portfolio’s sensitivity to parameters, and construct portfolio se-lection model with parameter sensitivity constraint in an effort to reduce the negative influence of parameter estimation error to optimal portfolio. The re-sulting model is a nonconvex quadratically constrained quadratic program (non-convex QCQP), which is NP-hard. By decomposing the nonconvex constraints, we propose an efficient branch-and-bound algorithm with QP relaxation as the bounding procedure. Numerical experiment indicates that our branch-and-bound algorithm is competitive with the commercial software BARON. To examine the model efficiency, we analyze the effect of the parameter robustness improvement and out-of-sample performance using backtesting strategy. The analysis results indicate that the portfolio model with parameter sensitivity control can construct efficient portfolio with good parameter robustness.Cardinality and minimum buy-in threshold constraints are often encountered in practical portfolio selection models due to the managerial and transaction cost considerations. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second-order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically con-strained quadratic (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effec-tiveness of the new MIQCQP reformulation. Since variance does not grasp all features of the risk of portfolios with asym-metry return, we introduce Value-at-Risk (VaR) constraint as a downside risk measure into mean-variance framework. VaR constraint is equivalent to proba-bilistic constraint and can be reduced to a group of mixed0-1linear constraints under finite discrete distributions. We first derive second-order cone program-ming (SOCP) relaxation and semidefinite programming (SDP) relaxation for the problem via a new Lagrangian decomposition scheme. We then give a mixed inte-ger quadratic programming (MIQP) reformulation of the problem and show that the continuous relaxation of the MIQP is exactly the SOCP relaxation. Com-putational results demonstrate the tightness of the new convex relaxations and the effectiveness of the new MIQP reformulation. Numerical results also suggest that the VaR constraint brings great computational challenge when there are large number of scenarios.Existing research results indicate that VaR based on scenarios or samples is not robust, while the parametric VaR sometimes exhibits large estimation error if the assumed distribution has large deviation from true distribution. Non-parametric VaR, on the other hand, is a robust estimator not relying on the distribution assumption. We propose portfolio selection models by incorporating nonparametric VaR into portfolio selection. Since the resulting problem is non-convex and hard to solve, we develop an alternative direction algorithm based on augmented Lagrangian method. To show the performance of the proposed model, we carry out empirical analysis on the statistic performance and portfolio behavior using American market data. The empirical results show that nonpara-metric VaR generally behaves robust and the corresponding portfolio selection model generates portfolio with good performance.It is well-known that the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlin-ear payoff structures. VaR is particularly suitable as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this thesis nonlinear portfolio selection models using approximate parametric VaR. More specifically, we use first-order and second- order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta-Gamma-normal and worst-case Delta-Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta-Gamma-normal VaR approximation performs the best in terms of balancing the trade-off between the approximation accuracy and computational efficiency.The thesis is organized as follows. In Chapter1, we briefly introduce the research background of portfolio selection problems and summarize the main con-tributions of the thesis. In Chapter2, we give an overview of the basic theory and research results of portfolio selection. Portfolio selection problems with fac-tor risk control is studied in Chapter3. In Chapter4, we discuss the parameter sensitivity and propose a portfolio selection problem with parameter sensitiv-ity control. Portfolio selection problems with cardinality and minimum buy-in threshold constraints are investigated in Chapter5, while mean-variance portfo-lio optimization model with VaR constraint is presented and analyzed in Chapter6. In Chapter7, the portfolio selection problem based on nonparametric VaR is studied. In Chapter8, we propose the nonlinear portfolio selection based on parametric VaR. We analyze the approximation accuracy of different parametric VaR and present empirical performance of different models. Finally, we sum-marize in Chapter9the results of the thesis and discuss some future research perspectives.